Zero field isotropic Muonium Kubo — Toyabe relaxation functions

Static zero field Gaussian Kubo — Toyabe relaxation functions for muons in isotropic muonium atoms are presented. That is, as with diamagnetic muons, an average of the spin dynamics of a muon in an isolated isotropic ground state muonium atom is taken over an isotropic Gaussian continuous classical local random magnetic field distribution. This motion approximates the exact quantal spin dynamics generated by the dipole-dipole interactions between the muonium atom and the surrounding nuclear spins associated with the site at which the muonium atom has stopped. Expressions are derived for triplet muonium only since, in general, singlet muonium is not observed. For normal nuclear spins and ground state muonium, the resulting relaxation functions are identical to the standard diamagnetic function (except for a shift in the time scale).     

Interaction of anomalous muonium with electronic excitations in Si studied by longitudinal field μSR

We report longitudinal muon spin relaxation measurements in Si doped with phosphorus below room temperature. The data can be described qualitatively in a model where bond-centered muonium is undergoing spin exchange interactions below 150 K. Above this temperature, charge state cycling becomes important.This work is partially supported by the National Sciences and Engineering Research Council of Canada.     

Theory for position of muonium in α-quartz and associated hyperfine interaction

The electronic structures and hyperfine interactions of muonium and hydrogen in -quartz are investigated by the unrestricted Hartree-Fock cluster procedure. The muonium is found to be trapped near the center of the line joining two silicon atoms. On including vibrational effects, the muon hyperfine constant comes out as 1.09 times that for free muonium, this ratio being larger than unity and smaller than for protons in trapped hydrogen, both features being in agreement with experiment.Briefly reported in Abstract at the American Physical Society meeting in New Orleans, March 1988. See Bull. Am. Phys. Soc. 33 (1988) 770.     

Hyperfine interactions of muonium in alpha-quartz

We present the properties of muonium which are connected with the presence of the muonium quadrupole moment and discuss the results of experimental studies of the quadrupole interactions of muonium with the alpha-quartz lattice.     

Atomic hydrogen and muonium in alkali halides

Atomic hydrogen can be trapped at interstitial and substitutional cation and anion sites in alkali halides. The geometrical structure of these defects was established by electron nuclear double resonance (ENDOR). From the analysis of the ENDOR spectra also detailed information was obtained on the electronic structure. In this article the major experimental and theoretical results for atomic hydrogen in several alkali halides are briefly reviewed. Special emphasis is given to the isotope effects upon replacing hydrogen by deuterium. The nature of the dynamical hyperfine and superhyperfine interactions is discussed. Its magnitude to be expected for muonium is estimated. Recent results on muonium centres are discussed on the basis of the knowledge about the hydrogen centres.     

Quantum diffusion of muonium in insulators

In this paper we review our recent experiments conducted at TRIUMF on muonium diffusion in alkali halides. First, the technique of longitudinal-field muonium spin relaxation (T ₁) due to nuclear hyperfine interaction, an indispensabletour de force for the present work. is described. It is demonstrated in KCl that the technique provides spectacular sensitivity for muonium diffusion as well as determining the average nuclear hyperfine coupling constant. The muonium hop rate shows a minimum (T ^*≃80 K) and steep increase with decreasing temperature. The result is compared with the current theory of quantum diffusion in non-metallic crystals. A few more sets of new data may be presented for other alkali halides. In addition, we show that muonium forms a delocalized state in NaCl as evidenced by a large change of the average nuclear hyperfine parameter. Related topics of local tunneling system may be briefly reviewed.     

Radiofrequency muon depolarization in the muonium + nuclear spin system

The effect of a considerable strengthening of muon depolarization in ALC resonance experiments was predicted for the muonium + nuclear spin system in the presence of a radiofrequency field. A mathematical approach was developed for obtaining analytic solutions that described the muon spin dynamics in ALC experiments, including a particular exact solution that contained much information about the system studied in fairly low magnetic radiofrequency fields. An analysis of these solutions and numerical calculations allowed us to comprehensively analyze muon depolarization patterns in a radiofrequency field. The results reveal the potential of muon depolarization strengthening for considerably increasing the sensitivity of experimental studies of muonium interactions with neighboring nuclear spins and for obtaining new spectroscopic information.     

Location of muonium and hydrogen in C60 fullerene and associated electronic structure and hyperfine properties

The unrestricted Hartree-Fock (UHF) procedure is used to investigate the locations, associated electronic structures and hyperfine interactions for muonium and hydrogen in C₆₀ fullerene. Our results indicate that from total energy considerations, in keeping with earlier investigations, the exohedral model has the lowest energy. However, the energies of the endohedral model involving the muonium (hydrogen) inside the fullerene and bonded to one of the carbon atoms, and of the muon at the center are found to be almost equal, contrary to earlier results. The hyperfine interaction constant for the endohedral site is in good agreement with that required to explain the lower observed muon spin-rotation (SR) frequency in the C₆₀-muonium system. The same appears to be the case for the exohedral model. However, there seems to be some uncertainty about the theoretical result in the latter case due to significant admixtures of higher spin states in the UHF wave-function. Additionally, in solid fullerene, the calculated location of the muonium for the exohedral model is such that it could be bonded to two fullerene molecules and therefore a muonium attached to a simple fullerene may not be representative of the exohedral state. This feature as well as the difficulty for the exohedral model of explaining the observed equality of the correlation times for relaxation effects associated with both SR and¹³C relaxation times in nuclear magnetic resonance (NMR) experiments suggests that the endohedral model for muonium cannot at present be ruled out as a viable model in favor of the exohedral model. Possible avenues for future investigations to resolve some of the problems for both exohedral and endohedral models are discussed. Results obtained for muonium at the center of fullerene are presented and compared to the features of the observed high frequency SR signal, and possible improvements in theory are discussed.     

Electronic structure and hyperfine interaction of muonium in semi-conductors

The Unrestricted Hartree-Fock self-consistent field cluster procedure is being utilized for first-principle investigations of the electronic structures and hyperfine interactions in normal and anomalous muonium states in semi-conductors. Our results for the total energy for the normal muonium state for a twenty-seven atom cluster in diamond, including the muonium and its neighboring atoms, show a minimum at the tetrahedral site and a maximum at the hexagonal site indicating that normal muonium is located in the tetrahedral region and avoids the hexagonal region. Using the calculated spin-density as a function of the position of muonium and carrying out averaging over the vibrational motion of the muon governed by the total energy curve obtained from our work, we have derived a muon hyperfine constant which is about 75% of that in free muonium, in good agreement with experiment. The natures of the total energy and spindensity curves permit us to draw conclusions regarding the origin of the observed trend in the hyperfine constants for normal muonium in diamond, silicon and germanium. The UHF cluster procedure is also applied to study a model of a muon in a positively charged environment for the anomalous muonium center in diamond. This model leads to a hyperfine interaction tensor with the observed feature of strong anisotropy but significantly weaker than experiment. The results obtained for this model indicate the importance for the anomalous muonium state with its relatively weak hyperfine interaction, of exchange polarization effects inherent in the UHF procedure.     

Anomalous muonium relaxation rates in quartz at high temperatures

Muonium spin relaxation measurements of the \alpha‐, \beta‐, and \gamma‐tridymite phases of quartz have been carried out over a temperature range from 300 to 1250 K. Anomalous relaxation rate increases are observed which may result either from resonance coupling between the other impurity ions and defects in the lattice and the diffusing muonium, or may result from phonon interactions with the muonium quadrupole moment. This revised version was published online in August 2006 with corrections to the Cover Date.     

Muonium

Muonium, the bound state of the two structureless leptons μ⁺ and e⁻, is an ideal system to test quantum electrodynamics and tiny contributions of other interactions. This paper reviews the latest state of the measurements and theories for the muonium hyper-fine structure in the ground state and the 1S–2S splitting in muonium. The results of these experiments are indispensable to interpret a more precise measurement of the anomalous magnetic moment of the muon presently underway. Furthermore, the search for muonium-antimuonium conversion is reported and discussed with respect to lepton number conservation.     

Determination of Easy Axis in a Chemical Ferromagnet, 4-(p-Chlorobenzylideneamino)–TEMPO (TEMPO=2,2,6,6-Tetramethylpiperidin-1-Oxyl)

The trapping sites for muon and muonium in ferromagnetic p-Cl–Ph–CH=N–TEMPO [(4-(p-chlorobenzylideneamino)–TEMPO (TEMPO = 2,6,6-tetramethyl-piperidin-1-yloxyl)] and the hyperfine interaction tensors for these sites are obtained using first-principles Unrestricted Hartree–Fock theory. The calculated hyperfine interactions are used to compare the calculated zero field muon spin rotation (μSR) frequencies for different choices for the easy axis and the observed frequency. It has been concluded that the two trapping centers that can best explain the observed μSR frequency are a trapped singlet muonium near the radical oxygen and a trapped muon site near the chlorine. The direction of the easy axis also is determined to be the b-axis of the monoclinic lattice. This direction for the easy axis is confirmed by determining the direction of the distributed magnetization in the molecular solid which leads to a minimum dipole–dipole interaction energy. The consequences of this agreement for the easy axis direction by two independent procedures are discussed. This revised version was published online in September 2006 with corrections to the Cover Date.     

Recent advances in positive muon spin rotation; muonium as a probe of fullerenes

In non-metallic solids the positive muon often forms paramagnetic muonium centers which are characterized by the hyperfine interactions of the unpaired electron with the positive muon and with the surrounding nuclear spins. The static and fluctuating components of these hyperfine interactions provide information on local molecular dynamics and local electronic structure. Some recent results on C₆₀ and related compounds are presented to illustrate this.     

Muonium in ice

Muonium has been studied in single crystals of H₂O and D₂O. Two-frequency precession in low transverse fields and a single zero-field oscillation indicate a small anisotropy of axial symmetry in the muonium hyperfine interaction. The anisotropy is shown to be the cause of the hitherto unexplained temperature independent contribution to muonium spin relaxation in polycrystalline samples. Relaxation rates for 99 K–263 K are reported for muonium in a single crystal of H₂O. Relaxation is attributed to electron-nuclear dipolar coupling of muonium to lattice protons, modulated by translational diffusion of muonium alongc-axis channels of the ice lattice. A simple model for H and Mu diffusion in ice is investigated.This work was supported by the Natural Sciences and Engineering Research Council of Canada through an Intermediate Energy Physics Project Grant.     

Field dependence of the longitudinal muon polarization for anomalous muonium

The constant muon polarization for anomalous muonium exhibits a peculiar field dependence which represents an easily measurable signature of anomalous muonium centers even in polycrystalline materials. Furthermore it can be used to extract information on the dynamical destruction of the state upon temperature variations and it might also be useful to investigate muonium in amorphous materials.This work has been partially supported by the Swiss National Science Foundation.     

μSR in diamond

It has recently become possible to take advantage of the interesting properties of diamond, as the synthesis of diamond has reached a point that material with impurities in the low the part per billion range and residual strain in the order of ten nano-radians has been realized. Hyperfine interactions have played a key role in the study of diamond and the emergence of novel applications. This relates to the characterization of defects and the development of engineered few spin systems. A particular aspect of the defect studies is the elusive hydrogen defect. In μSR studies, muonium is considered a light isotope of hydrogen with very similar chemical properties, but with very interestingly different dynamical properties, due to its much lighter mass. It offers a unique opportunity to study the behaviour of hydrogen in diamond at very low concentrations. The studies have revealed details of the elementary muonium atom as well as a molecule involving muonium in the diamond lattice. The dynamics of the muonium, which include quantum diffusion and ionization have also been studied. This contribution reviews μSR in diamond in the context of diamond as a modern material hosting advanced applications.     

Muons and muonium in cryocrystals

We present a collection of measurements of the muon and muonium asymmetries and relaxation parameters in cryocrystals of N₂, CO, Xe,¹³⁶Xe and Ne as functions of temperature. Generally, the fractions of the two species can be attributed to a competition between the formation of muonium or a diamagnetic species, where processes involving transport of spur electrons are important.     

Muonium defect centers in semiconductors

After a brief summary of the properties of the muonium defect centers observed in the elemental group IV semiconductors, the status of studies of muonium centers in semiconductors at the time of the last μSR conference in 1983 will be compared with what is currently known. With the introduction of new experimental techniques, such as high-transverse-field μSR and level-crossing spectroscopy, many new results are or soon will be available on muonium centers. These, combined with new theoretical studies, should lead to rapidly increased insight into a subject which has been both puzzling and resistant to clarification.     

Hydrogen and muonium in silicon

First principles calculations of the properties of hydrogen and muonium in silicon are presented. H⁺ and H⁻ are shown to have definite preferences for bond-centred and tetrahedral interstitial sites respectively whereas H⁰ (or a muon) is shown to be stable at two sites with almost equal energies, the bond-centred and antibonding sites. The structures of normal and isotropic muonium are discussed. In contrast to common belief the tetrahedral site is shown to be unstable with the muon moving spontaneously towards one of the neighbouring silicon atoms. The barrier to motion between equivalent antibonding sites is low suggesting that the normal muonium signal is isotropic because of motional averaging, not due to the symmetry of a well defined equilibrium site.